Finding k for One Positive and One Negative Root in a Quadratic Equation

Finding 'k' for One Positive and One Negative Root in a Quadratic Equation

This problem involves the quadratic equation kxﾲ -(k+3)x + 2k + 9 = 0, where 'k' is a real number. We aim to find the values of 'k' so that this equation has one positive and one negative real root.

1. Product of Roots:

Vieta's formulas provide a shortcut to finding the product of a quadratic equation's roots. For a quadratic in the standard form axﾲ + bx + c = 0, Vieta's formulas state that the product of the roots is equal to c/a.

In our case, a = k and c = 2k + 9. Therefore, the product (P) of the roots is:

P = (2k + 9)/k

2. Discriminant Analysis:

The discriminant of a quadratic equation (denoted as 'D') provides information about the nature of its roots:

• D > 0: Two distinct real roots- D = 0: One real root (a repeated root)- D < 0: Two complex roots

For our equation (kxﾲ -(k+3)x + 2k + 9 = 0), the discriminant is:

D = (-(k+3))ﾲ - 4(k)(2k + 9)D = (kﾲ + 6k + 9) - 8kﾲ - 72kD = -7kﾲ - 66k + 9

To have one positive and one negative root, the discriminant must be positive (D > 0). This leads to the inequality:

-7kﾲ - 66k + 9 > 0

3. Solving the Inequality:

Solving this quadratic inequality will provide the range of 'k' values that satisfy the condition of one positive and one negative root. You can solve this inequality using factoring, the quadratic formula, or graphical methods. The solution will be a range of 'k' values that make the discriminant positive, ensuring the quadratic equation has the desired root characteristics.